Average Error: 14.7 → 2.3
Time: 19.9s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \mathsf{fma}\left(0.3333333333333333, \left(\pi \cdot \ell\right) \cdot F, \mathsf{fma}\left(0.0021164021164021165, F \cdot \left({\ell}^{5} \cdot {\pi}^{5}\right), 0.022222222222222223 \cdot \left(F \cdot {\left(\pi \cdot \ell\right)}^{3}\right)\right)\right)}}{F} \]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \mathsf{fma}\left(0.3333333333333333, \left(\pi \cdot \ell\right) \cdot F, \mathsf{fma}\left(0.0021164021164021165, F \cdot \left({\ell}^{5} \cdot {\pi}^{5}\right), 0.022222222222222223 \cdot \left(F \cdot {\left(\pi \cdot \ell\right)}^{3}\right)\right)\right)}}{F}

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))
↓
(FPCore (F l)
 :precision binary64
 (-
  (* PI l)
  (/
   (/
    1.0
    (-
     (/ F (* PI l))
     (fma
      0.3333333333333333
      (* (* PI l) F)
      (fma
       0.0021164021164021165
       (* F (* (pow l 5.0) (pow PI 5.0)))
       (* 0.022222222222222223 (* F (pow (* PI l) 3.0)))))))
   F)))
double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan(((double) M_PI) * l));
}
↓
double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / ((F / (((double) M_PI) * l)) - fma(0.3333333333333333, ((((double) M_PI) * l) * F), fma(0.0021164021164021165, (F * (pow(l, 5.0) * pow(((double) M_PI), 5.0))), (0.022222222222222223 * (F * pow((((double) M_PI) * l), 3.0))))))) / F);
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 14.7

    \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]

  2. Simplified14.5

    \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

  3. Applied associate-/r*_binary6411.3

    \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

  4. Applied clear-num_binary6411.3

    \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}}}{F} \]

  5. Taylor expanded in l around 0 2.4

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\color{blue}{\frac{F}{\pi \cdot \ell} - \left(0.3333333333333333 \cdot \left(\pi \cdot \left(F \cdot \ell\right)\right) + \left(0.0021164021164021165 \cdot \left({\pi}^{5} \cdot \left({\ell}^{5} \cdot F\right)\right) + 0.022222222222222223 \cdot \left({\pi}^{3} \cdot \left({\ell}^{3} \cdot F\right)\right)\right)\right)}}}{F} \]

  6. Applied *-un-lft-identity_binary642.4

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(0.3333333333333333 \cdot \left(\pi \cdot \left(F \cdot \ell\right)\right) + \left(0.0021164021164021165 \cdot \left({\pi}^{5} \cdot \left({\ell}^{5} \cdot F\right)\right) + 0.022222222222222223 \cdot \left({\pi}^{3} \cdot \left({\ell}^{3} \cdot F\right)\right)\right)\right)}}{\color{blue}{1 \cdot F}} \]

  7. Applied associate-/r*_binary642.4

    \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(0.3333333333333333 \cdot \left(\pi \cdot \left(F \cdot \ell\right)\right) + \left(0.0021164021164021165 \cdot \left({\pi}^{5} \cdot \left({\ell}^{5} \cdot F\right)\right) + 0.022222222222222223 \cdot \left({\pi}^{3} \cdot \left({\ell}^{3} \cdot F\right)\right)\right)\right)}}{1}}{F}} \]

  8. Simplified2.3

    \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{1}{\frac{F}{\ell \cdot \pi} - \mathsf{fma}\left(0.3333333333333333, F \cdot \left(\ell \cdot \pi\right), \mathsf{fma}\left(0.0021164021164021165, F \cdot \left({\ell}^{5} \cdot {\pi}^{5}\right), 0.022222222222222223 \cdot \left(F \cdot {\left(\ell \cdot \pi\right)}^{3}\right)\right)\right)}}}{F} \]

  9. Final simplification2.3

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \mathsf{fma}\left(0.3333333333333333, \left(\pi \cdot \ell\right) \cdot F, \mathsf{fma}\left(0.0021164021164021165, F \cdot \left({\ell}^{5} \cdot {\pi}^{5}\right), 0.022222222222222223 \cdot \left(F \cdot {\left(\pi \cdot \ell\right)}^{3}\right)\right)\right)}}{F} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))